Many modern wireless and other communication systems encode digital data for transmission over communication paths. One type of coding mechanism involves the use of polar codes, which are used to encode data words into encoded data words that are then mapped into symbols for transmission. Polar codes can be used to achieve capacity of binary-input memoryless symmetric channels, meaning the polar codes allow the full capacity of a channel to be utilized. Polar codes allow this to be achieved using low-complexity encoding and decoding algorithms.
Recently, various attempts have been made to use communication channels more efficiently based on higher-order modulation techniques, such as multilevel quadrature amplitude modulation (M-QAM). However, extending the use of polar codes to systems that use higher-order modulation techniques is not a simple task. For example, output bits from a binary polar code often experience different effective channels because of their mapping to higher-order symbols. As a result, standard techniques for polar code design are not generally applicable to systems using higher-order modulation techniques.